The difference between the simple interest and the compound interest, compounded annually, on a certain sum of money for 2 years at 10% per annum is $₹407$. Find the sum [rounded off to the nearest integer].

Question

Prepp Team · Accepted Answer

This problem requires us to find the principal sum (the initial amount of money) based on the difference between simple interest and compound interest calculated over two years at a specific interest rate.

Understanding Simple Interest (SI) and Compound Interest (CI)

Simple Interest (SI): Interest calculated only on the principal amount for the entire duration.
Compound Interest (CI): Interest calculated on the principal amount plus the accumulated interest from previous periods. It is compounded annually in this case.

Calculating the Difference Between CI and SI

For a principal amount '$P$', rate '$R\%$' per annum, and time '$T$' years, the difference between compound interest (compounded annually) and simple interest for 2 years is given by the formula:

$$ \text{Difference} = P \times \left(\frac{R}{100}\right)^2 $$

Applying the Formula to Find the Sum

We are given the following information:

Difference between CI and SI = $₹407$
Rate of interest (R) = $10\%$ per annum
Time = 2 years

Let the principal sum be '$P$'. Plugging the values into the formula:

$$ ₹407 = P \times \left(\frac{10}{100}\right)^2 $$

First, simplify the fraction for the rate:

$$ \frac{10}{100} = \frac{1}{10} $$

Now, substitute this back into the equation:

$$ ₹407 = P \times \left(\frac{1}{10}\right)^2 $$ $$ ₹407 = P \times \frac{1}{100} $$

To find the principal sum '$P$', rearrange the equation:

$$ P = ₹407 \times 100 $$ $$ P = ₹40,700 $$

The calculated sum is $₹40,700$. Since the question asks for the sum rounded off to the nearest integer, and our result is already an integer, no rounding is necessary.